First, I would like to describe the relation between the embedding of an étale algebra into a central simple algebra and the embedding of a twisted root datum into a reductive group. In the second part, I will focus on the embedding of a twisted root datum into a reductive group over local fields. In this case, the Tits index determines the existence of the embedding. I would also like to explain the obstruction to the local-global principle for the embedding problem and provide an example when the local-global principle for the embedding problems fail.